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When the cable is stretched tight TO is large compared with w. Therefore the higher terms of the series may be neglected and equation (19) be put in the following approximate form.
Hence the increase in length due to sagging is  ^r2D3,
24:    jt o
1.   A perfectly flexible cord hangs over two smooth pegs, with its ends hanging freely, while its central part hangs in the form of a catenary.    If the two pegs are on the same level and at a distance D apart, show that the total length of the string must not be less than De, in order that equilibrium shall be possible, where e is the natural logarithmic base.
2.   In the preceding problem show that the ends of the cord will be on the x-axis.
3.   Supposing that a telegraph wire cannot sustain more than the weight of one mile of its own length, find the least and the greatest sag allowable in a line where there are 20 poles to the mile.
4.   Find the actual length of the wire per mile of the line in the preceding problem.
6.   The width of a river is measured by stretching a tape over it. The middle point of the tape touches the surface of the water while the
ends are at a height // from the surface.   If the tape reads S, show that
the width of the river is approximately \ /  - --
V       J!t
' 6. Show that the cost of wire and posts of a telegraph line is minimum if the cost of the posts is twice that of the additional length of wire required by sagging. The posts are supposed to be evenly spaced and large in number.
7.   A uniform cable which weighs 100 tons is suspended between two points, 500 feet apart, in the same horizontal line.   The lowest point of the cable is 40 feet below the points of support.   Find the smallest and the greatest values of the tensile force.
8.   In the preceding problem find the length of the cable.
66.  Friction Belts.  The flexible cord AB, Fig. 44, is in equilibrium under the action of three forces, namely, To